1. Observing Quantum Monodromy
An Energy-Momentum Map Built From Experimentally-Determined Level Energies Obtained from the ν7 Far-Infrared Band System of NCNCS
Dennis W. Tokaryk, Stephen C. Ross
Department of Physics and Centre for Laser, Atomic, and Molecular Sciences
University of New Brunswick, Fredericton, NB Canada
Brenda P. Winnewisser, Manfred Winnewisser, Frank C. De Lucia
Department of Physics, The Ohio State University, Columbus, OH USA
Brant E. Billinghurst
Canadian Light Source, Inc., Saskatoon, SK Canada
MF09 – 71st International Symposium on Molecular Spectroscopy University of Illinois June 20, Slide 1

2. Monodromy – you’ve seen it before!
A simple example from complex analysis: let z = x + iy = r eiθ.
Then the function,
ln(z) = ln(r) + iθ,
has a singularity at z = 0 + i0 = 0 eiθ! (because ln(0) is undefined.)
Re(z)
Im(z) z y r θ x
MF09 – 71st International Symposium on Molecular Spectroscopy University of Illinois June 20, Slide 2

3. Monodromy – you’ve seen it before!
Make a circuit around the singularity, evaluating ln(z) en route:
At the start of the circuit θ = 0, so ln(z) = ln(r0) + i0 = ln(r0).
Re(z)
Im(z)
Start here
ln(z) = ln(r0) z θ r0
MF09 – 71st International Symposium on Molecular Spectroscopy University of Illinois June 20, Slide 3

4. Monodromy – you’ve seen it before!
At the end of the circuit, θ = 2πi, so ln(z) = ln(r0) + 2πi.
The value of ln(z) is not single-valued:
- the value depends on: how we got to z (i.e. upon its history).
Im(z) θ
Re(z) r0 z
End here
ln(z)
= ln(r0) + 2πi
MF09 – 71st International Symposium on Molecular Spectroscopy University of Illinois June 20, Slide 4

5. quasi-linear molecules
Chemistry
Molecules
(synthesis)
Physics/
Technology
Spectroscopic
data
Quantum monodromy
as seen in
quasi-linear molecules
Theoretical
spectroscopy
GSRB Hamiltonian
Topology of
the phase-space
surfaces of
constant E, J
Mathematics
MF09 – 71st International Symposium on Molecular Spectroscopy University of Illinois June 20, Slide 5

6. Timeline of LAM theoretical and experimental spectroscopy
Year
1970
The Vibration-Rotation Problem in Triatomic Molecules Allowing for a Large-Amplitude Bending Vibration, HBJ = Hougen, Bunker and Johns, J. Mol. Spectrosc. 34, 136 (1970)
The effective rotation-bending Hamiltonian of a triatomic molecule, and its application to extreme centrifugal distortion in the water molecule, Hoy and Bunker, J. Mol. Spectrosc. 52, 439 (1974)
Semi-rigid bender (SRB) Hamiltonian by Bunker and Landsberg, J. Mol. Spectrosc. 67, 374 (1977)
1980
A reinterpretation of the CH2 photoelectron spectrum, Sears and Bunker, J. Chem. Phys. 79, 5265 (1983)
Analysis of the laser photoelectron spectrum of CH2, Bunker and Sears, J. Chem. Physics 83, 4866 (1985): We NOW know that an energy-momentum map of their calculated rotation-bending energies in their Table V shows quantum monodromy!
1990
Morse Oscillator Rigid Bender Internal Dynamics [MORBID], Jensen, J. Mol. Spectrosc. 128, 478 (1988).
OCCCS, NCNCS, NCNCO, and NCNNN as Semirigid Benders, [Hamiltonian now called: Generalized Semi-Rigid Bender (GSRB) ], Ross, J. Mol. Spectrosc. 132, 48 (1988)
Experimental confirmation of quantum monodromy: The millimeter wave spectrum of cyanogen isothiocyanate NCNCS, B. and M. Winnewisser, Medvedev, Behnke, DeLucia, Ross and Koput, PRL, (2005)
2000
The hidden kernel of molecular quasi-linearity: Quantum monodromy, M. and B. Winnewisser, Medvedev, DeLucia, Ross, Bates, J. Mol. Structure 798, 1-26 (2006)
Analysis of the FASSST rotational spectrum of NCNCS in view of quantum monodromy, B. and M. Winnewisser, Medvedev, DeLucia, Ross and Koput, PCCP, 12, 8158 ( 2010)
2010
Pursuit of quantum monodromy in the far-infrared and mid-infrared spectra of NCNCS using synchrotron radiation, M. and B. Winnewisser, DeLucia, Tokaryk, Ross and Billinghurst, PCCP, 16, (2014)
MF09 – 71st International Symposium on Molecular Spectroscopy University of Illinois June 20, Slide 6

7. Timeline of Mathematics leading to Monodromy in Molecules
Year
Timeline of Mathematics leading to Monodromy in Molecules
1970
Geometrical obstructions to global action-angle variables, Nekhoroshev, Trans. Moskow Math. Soc. 26, 180 (1972)
On global action-angle coordinates, Duistermaat, Comm. Pure and Appl. Math. 33, 687 (1980)
1980
The quantum mechanical spherical pendulum, Cushman and Duistermaat, Bull. Am. Math. Soc. 19, 475 (1988)
Monodromy in the champagne bottle, Bates, J. Appl. Math. and Physics 42, 837 (1991)
Classical energy-momentum map
1990
Quantum states in a champagne bottle, Child, J. Phys. A: Math. Gen. 31, 657 (1998)
Energy-momentum map of discrete quantum levels
Quantum monodromy in the spectrum of HOH and other systems: new insight into the level structure of quasi-linear molecules, Child, Weston and Tennyson, Mol. Phys. 96, 371 (1999)
2000
Monodromy in the water molecule, Zobov, Shirin, Polyansky, Tennyson, Coheur, Bernath, Chemical physics letters, 414, 193 (2005): Comparison of high temperature HOH data with predictions
Hamiltonian monodromy as lattice defect, Zhilinskii, Topology in condensed matter, Springer series: Vol. 150, 186 (2006)
2010
Quantum monodromy and molecular spectroscopy, Child, Contemporary Physics 55, 212 (2014)
MF09 – 71st International Symposium on Molecular Spectroscopy University of Illinois June 20, Slide 7

8. Classical motion in the champagne-bottle potential
The classical motion was studied in 1991 by Larry Bates.
(Bates, J. Ap. Math. Physics 42 (1991) 837)
He made an Energy-Momentum map and identified the ‘critical points’ at which the radial momentum is zero. These form the parabolic curve + the critical point (k = 0, e = the energy of the top of the potential energy hump) shown here:
k: angular momentum
e: total energy
Critical points
Energy-Momentum map
MF09 – 71st International Symposium on Molecular Spectroscopy University of Illinois June 20, Slide 8

9. Classical motion in the champagne-bottle potential
The classical action variable for radial motion is,
Ir ~ the frequency of a particle’s radial oscillation
(for the given energy e and angular momentum k).
• Evaluate Ir at a starting point
on the loop shown on the
energy-momentum map.
• Move a small distance around
the loop, re-evaluate Ir.
• Continue all the way around
the loop,
- calculate Ir at each point. • Result: Ir has a different value
when you return to the starting
point!
k: angular momentum
e: total energy Ir
MF09 – 71st International Symposium on Molecular Spectroscopy University of Illinois June 20, Slide 9

10. Classical motion in the champagne-bottle potential
Our classical action
experiences a branch cut as shown!
So, the vibrational periodicity Ir represents will not smoothly change when crossing over this line.
Branch cut
k: angular momentum
e: total energy
MF09 – 71st International Symposium on Molecular Spectroscopy University of Illinois June 20, Slide 10

11. Monodromy! Monodromy! Larry Bates (University of Calgary) GSRB:
Quantum Hamiltonian
Monodromy in the
champagne bottle
J. Appl. Math. and
Physics 42, 837 (1991) 2 1
µρρJρ2 2 1
µzzJz2
H = +
V(ρ) +
Many more terms +
= Tvib Tz-rot +
Classical Hamiltonian
Many more terms + pR 2 j 2
H = + +
V(R) 2m
2mR2
GSRB contains the essential core which leads to…
= Tvib + Tz-rot +
Monodromy!
Monodromy!
• GSRB already accounted for Monodromy before we even knew it existed!
MF09 – 71st International Symposium on Molecular Spectroscopy University of Illinois June 20, Slide 11

12. Quantum motion in the champagne-bottle potential
Mark Child quantized the classical problem, and made an energy-momentum map of discrete quantum levels.
(Child, J. Phys. A: Math. Gen. 31 (1998) 657; Contemporary Physics 55 (2014) 212)
quantum level of energy E and angular momentum k.
line of constant radial vibration
v = 1
v = 0
MF09 – 71st International Symposium on Molecular Spectroscopy University of Illinois June 20, Slide 12

13. a-type sub-bands b-type sub-bands b-type sub-bands
Spectra of the ν7 low-frequency high-amplitude bending mode and of the
of the ν3 hybrid band stretching mode, taken at the CLS.
~68 a-type sub-bands
of the ν7 mode
(For experimental details, see
Winnewisser et al,
PCCP Perspectives 16, (2014 ) )
a-type sub-bands
b-type sub-bands
b-type sub-bands
MF09 – 71st International Symposium on Molecular Spectroscopy University of Illinois June 20, Slide 13

14. The energy-momentum map
How we connected Ka stacks and
different vibrational levels
with data from the CLS
Up to
Ka = 10
Ka = 10
Ka = 12
Ka = 12
MF09 – 71st International Symposium on Molecular Spectroscopy University of Illinois June 20, Slide 14

15. The NCNCS energy-momentum map showing the quantum lattice for the ν7 in-plane large amplitude bending mode
Small Dots: show J = Ka levels calculated from the analysis of the pure rotational spectrum only
Centers of the Red Circles: show levels measured directly from the assignment of the FIR spectrum
MF09 – 71st International Symposium on Molecular Spectroscopy University of Illinois June 20, Slide 15

16. Analysis of the laser photo-electron spectrum of CH2, Bunker and Sears,
J. Chem. Phys (1985)
We NOW know that an energy- momentum map of the calculated rotation-bending energies in their Table V shows quantum monodromy
MF09 – 71st International Symposium on Molecular Spectroscopy University of Illinois June 20, Slide 16

17. Monodromy plot for water
Child, Contemporary Physics, (2014)
with data from Zobov et. al., (2005).
MF09 – 71st International Symposium on Molecular Spectroscopy University of Illinois June 20, Slide 17

18. • We determined precise relative energies for the lowest seven
The take-home part…
• We determined precise relative energies for the lowest seven
vibrational levels of the ν7 mode of NCNCS: for Ka up to (a maximum of) 12.
• Our energy-momentum map contains all of the structure due
to quantum monodromy expected for a quasi-linear
molecule.
• Stephen Ross’ GSRB Hamiltonian proved excellent at
predicting the positions of these energy levels, even though
only pure rotational spectra were included in the fitting.
• If you are working on a new quasi-linear molecule, an
energy-momentum map will be a helpful aid to determining
the height of the barrier to linearity, since the general
structure of the map will be the same for every such
molecule.
MF09 – 71st International Symposium on Molecular Spectroscopy University of Illinois June 20, Slide 18

19. • Natural Sciences and Engineering Research Council of Canada (NSERC)
Thanks
• Natural Sciences and Engineering Research Council of
Canada (NSERC)
– Discovery grants for Drs. Ross and Tokaryk
• Damien Forthomme and Colin Sonnichsen for help with the
experiments
• Staff at the CLS for technical support and accommodation
during the experiments.
MF09 – 71st International Symposium on Molecular Spectroscopy University of Illinois June 20, Slide 19

20. Classical motion in the champagne-bottle potential
Mark Child explains why not.
(Child, J. Phys. A: Math. Gen. 31 (1998) 657; Contemporary Physics 55 (2014) 212)
For small values of k > 0, between successive radial maxima, Δθ > 0, but very small.
.. but now, for the same initial conditions, Δθ ~ +π, and direction of rotation reverses!
Child shows that the derivative of the action wrt k is negative if k>0, and positive if k<0…
discontinuous at k = 0!
won’t depend on k (same radial vibrational frequency for all small k, positive or negative.)
MF09 – 71st International Symposium on Molecular Spectroscopy University of Illinois June 20, Slide 20

21. MF09 – 71st International Symposium on Molecular Spectroscopy University of Illinois June 20, Slide 21

22. MF09 – 71st International Symposium on Molecular Spectroscopy University of Illinois June 20, Slide 22

23. Quantum motion in the champagne-bottle potential
What did our game mean??
That the quantum numbers v and k, which we took as valid below the monodromy point, were not universal.
New quantum numbers of vibration and angular momentum are necessarily required above the monodromy point.
MF09 – 71st International Symposium on Molecular Spectroscopy University of Illinois June 20, Slide 23

24. Quantum motion in the champagne-bottle potential
Rules of the game:
Define a closed path γ through a set of quantum levels circling the monodromy point.
From each point, increase k by one unit, then increase v by one unit. Make a closed trapezium.
The base of the previous trapezium should not make a drastic change of orientation from the previous one.
Check the return point – once again, your horse turns into a cow!
MF09 – 71st International Symposium on Molecular Spectroscopy University of Illinois June 20, Slide 24